Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61800 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(20,26)(21,29)(23,25)(27,34)(30,31)(32,33);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,20,24,26)(21,30,32,27)(22,25,28,23)(29,34,33,31);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,21,24,32)(20,27,26,30)(22,31,28,34)(23,29,25,33);(5,13)(8,11)(12,18)(15,17)(19,22,24,28)(20,23,26,25)(21,31,32,34)(27,29,30,33);(4,10)(8,11)(9,16)(15,17)(19,23,24,25)(20,28,26,22)(21,29,32,33)(27,34,30,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485172);